Automorphic formsand

even unimodular lattices

Kneser neighbors of Niemeier lattices

Gaëtan Chenevier1 & Jean Lannes2

October 7, 2016

1Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128Palaiseau Cedex, France (before 10/2014). Université Paris-Sud, Département demathématiques, Bâtiment 425, 91405 Orsay Cedex (after 10/2014).

2Université Paris 7, UFR de Mathématiques, bâtiment Sophie Germain, avenuede France, 75013 Paris, France.

Preface

Automorphic forms are functions defined on adèle groups, derived fromharmonic analysis, whose theory represents a far-reaching generalization ofthe one of modular forms. The famous functoriality conjecture by Langlandspredicts unexpected connections between automorphic forms associated toquite different groups. Recent advances confirm a part of these general con-jectures, as well as their refinements by Arthur, for classical groups. Thetechnicality of the proofs is formidable but in contrast the statements arefascinating by their extreme beauty, their wide range of applications and tosome extent their simplicity. Our aim in this memoir is to reconsider someproblems of classical origin, of number theory and of the theory of quadraticforms, from the angle of these recent results.

A special case, in which the Langlands conjectures keep however theirwhole flavour, while freed from numerous difficulties existing in general, isthe case where one restricts oneself to the study of automorphic forms un-ramified at each prime. An alternative terminology is level 1 automorphicforms. When one deals with classical modular forms or Siegel’s modularforms, historical examples if there were any of automorphic forms, this as-sumption means that one considers only forms which are modular for thegroups SL2(Z) or Sp2g(Z), and not for general congruence subgroups.

What is the interest of the case of level 1 automorphic forms is not only thesimplifications it provides, it is also very appealing to the arithmetician bythe mix of rarity and elegance of the examples (again think of the theory ofmodular forms for SL2(Z)). Furthermore it is linked, sometimes very directly,sometimes much less so, sometimes only conjecturally, to the objects of al-gebraic geometry (varieties, stacks) which are both proper and smooth overthe ring Z of integers and even to motives over the rationals with everywheregood reduction, objects as fascinating as mysterious.

In this work we aim to study the conjectures by Arthur and Langlands inthis context of level 1 automorphic forms, to give precise formulations of thestatements arising from the work of Arthur in this framework, and to illus-trate the latter by examples which are more specific but particularly spicy.We will confront also Arthur’s results to the ones derived from certain moreclassical constructions, theta series, which put within reach numerous exam-ples. Some of these constructions appear to be even richer, as we discovered,when they are combined with the triality principle. Let us emphasize thatwe want to work if possible with groups of large rank, as they best revealthe riches of the general phenomena, and to move away from the classical

i

examples given by “small" groups such as GL2 which have already been thesubject of an extensive literature.

Our illustrations will mainly concern the theory of quadratic forms overZ which are non-degenerate and positive definite, in other words the the-ory of even (integral) Euclidean lattices whose determinant is 1 or 2. Thisassumption about the determinant exactly means that the associated projec-tive quadric is smooth over Z, in which case the associated special orthogonalgroup is smooth (and even reductive) over Z. In the dimensions (less thanor equal to 25) for which these objects are classified, the concrete problemwe are going to address is the determination, for each prime number p, ofthe number of p-neighborhoods in Kneser’s sense, between the classes of suchobjects. We will call it the p-neighbor problem.

The p-neighbor problem can be quite elementarily approached: this is thepoint of view that we choosed to follow in the introduction (Chapter I) andalso in the organization of the memoir where it will serve as a connectingthread. This will also make it possible to begin by exposing the rich andfascinating history of the subject, and to bring out some simple but stri-king statements which are consequences of our results (dimension 16 case,determination of the p-neighborhood graphs in dimension 24, affirmation ofthe conjecture by Nebe-Venkov about the linear combinations of higher genustheta series of Niemeier lattices. . . ). However, we think it helpful to explainupstream our original motivation which was to test Arthur’s results in acontext both concrete and of high dimension, a motivation which will onlybe lying in the background in the introduction.

In the rest of this preface, we will explain the place of the p-neighborproblem in the general landscape of Langlands’ conjectures, or even motives,as well as the line of thoughts that led us to this problem. We hope thatthis enlightenment (or darkening depending upon the viewpoint!) will arousethe interest of the readers who are maybe less sensitive to the appeal of thetheory of Euclidean lattices. In any case this passage will be inevitable inorder to understand the ideas of the solution we propose of the p-neighborproblem, which uses in a crucial way the aforementioned recent developments.This apparent disproportion between the sophistication of methods and theelementary aspect of the p-neighbor problem is one of the charms of it.

The plan will be as follows. First, we will come back in a more preciseway on the notion of level 1 automorphic form (studied in Chapter IV ofthe memoir). After having discussed a few examples, we will present brieflyLanglands’ conjectures, emphasizing a statement that we call the Arthur-Langlands conjecture (Chapters VI & VIII). We will explain how Langlands

ii

and Arthur motivate this conjecture by means of a certain hypotheticalgroup, the Langlands group of Z, that we denote by LZ. When one spe-cializes the statements to algebraic automorphic forms, one may to a largeextent replace LZ with the absolute Galois group of Q. Then we will bein a position to provide the enlightenments we promised above, and also toglimpse some of the problems still to be solved once Arthur’s results havebeen “put in the engine”.

Automorphic forms of level 1

Let us fix an algebraic group (scheme) G defined and reductive over thering Z of integers. This means that G is connected, smooth over Z, and thatits reduction modulo p is reductive over Z/pZ for each prime p. The mostimportant examples are GLn and the famous Chevalley groups, or the groupswhich are isogenous to them such as PGLn, but other examples will play arole in the sequel.

The adele group G(A) is a locally compact topological group in a naturalway, restricted product of the real Lie group G(R) and of the p-adic Liegroups G(Qp) over all primes p; the subgroup G(Q) is discrete in G(A). Wedenote by Z the neutral component of the center ofG(R) (so Z equals 1 ifG issemisimple). The homogeneous spaceG(Q)\G(A)/Z is equipped with a finiteG(A)-invariant Borel measure. A central question is to describe the Hilbertspace L2(G(Q)\G(A)/Z) of square integrable automorphic forms of G, seenas a unitary representation of G(A) for the right translations. According toour objectives, we shall limit ourselves to considering the subspace

A2(G) = L2(G(Q)\G(A)/Z ·G(Z))

of automorphic forms of level 1, which is nothing else than the subspaceof G(Z)-invariants of L2(G(Q)\G(A)/Z). This is a Hilbert space equippedwith a natural unitary representation of G(R), and for each prime p, with anaction of the convolution ring

Hp(G) = Z[G(Zp)\G(Qp)/G(Zp)],

whose elements are the Hecke operators at p. The aim is to describe A2(G)equipped with these commuting actions of G(R) and of the commutative ringH(G) :=

⊗p Hp(G).

Denote by Π(G) the set of isomorphism classes of objects of the formπ∞ ⊗ πf , with π∞ an irreducible unitary representation of G(R) and πf aone-dimensional complex representation of the ring H(G). Such a πf may

iii

equally be viewed as a collection of ring homomorphisms1 πp : Hp(G) →C; we also talk about systems of eigenvalues of Hecke operators. Denotemoreover by m(π) the multiplicity of π as a subrepresentation of A2(G); it isfinite according to Harish-Chandra. A discrete automorphic representationof G is an element π of Π(G) with m(π) 6= 0. Let us denote at last byΠdisc(G) ⊂ Π(G) the subset of those representations. For general reasons, wemay write

(1) A2(G) = A2disc(G)

⊥⊕A2

cont(G) with A2disc(G) '

⊥⊕

π∈Πdisc(G)m(π) π.

The space A2disc(G) contains the subspace A2

cusp(G) of cuspforms, whosedefinition is a natural generalization of the one of cuspidal modular form. Wedenote by Πcusp(G) ⊂ Πdisc(G) the subset of elements appearing in A2

cusp(G).The description of the subsets Πcusp(G) ⊂ Πdisc(G) of Π(G) is the heart ofthe problem. Indeed, we know since Langlands how to describe the contin-uous part A2

cont(G) in terms of the A2cusp(L) where L runs through the Levi

subgroups of all the proper parabolic subgroups of G defined over Z. We willnot be interested in A2

cont(G) in this memoir.

Two examples

The representations π in Πdisc(G) have very different concrete manifes-tations depending on the nature of their Archimedean component π∞. IfU is an arbitrary irreducible unitary representation of G(R), and if we setAU(G) := HomG(R)(U,A

2(G)), then we have

AU(G) = HomG(R)(U,A2disc(G)) '

⊕{π ∈ Πdisc(G) |π∞'U}

m(π) πf

This is an H(G)-module in an obvious way, and a finite dimensional complexvector space according to Harish-Chandra. It is equivalent to describe thewhole of Πdisc(G) or the H(G)-modules AU(G) when U runs through theunitary dual of G(R).

In order to illustrate these notions, it is instructive to specify them in thespecial case of the group G = PGL2.2 If U is a discrete series representation,

1We do not follow here the tradition according to which πp rather denotes the isomor-phism class of the (irreducible) C[G(Qp)]-submodule of L2(G(Q)\G(A)/Z) generated byan arbitrary nonzero element of π. The difference is however artificial, as it is a well-kwownconsequence of the commutativity of Hp(G) that the two definitions exactly contain thesame information.

2Following our definitions, we have a canonical isomorphism A2(PGLn)∼→ A2(GLn).

iv

say with lowest weight the (even) integer k > 0, then AU(G) naturally iden-tifies with the space of cuspforms of weight k for SL2(Z) equipped with theaction of the standard Hecke operators on the latter. If U := Us is a princi-pal or complementary series, parameterized in the usual way by an elements ∈ iR ∪ [0, 1[, then AUs(G) identifies with the Hecke-module of cuspidalMaass forms whose eigenvalue is 1−s2

4for the action of the Laplace opera-

tor on the Poincaré upper half-plane. Contrary to the previous case, thesespaces are very mysterious: Selberg has proved3 AUs(G) = 0 for s > 0, butwe do not know any single exact value of s such that AUs(G) is nonzero, orwhether the latter space can be of dimension > 1. Finally, the unique unitaryrepresentation left of PGL2(R) is, according to Bargmann, the trivial rep-resentation 1, and we obviously have dimA1(G) = 1 (consider the constantfunctions).

Let us now discuss the example which will be of a great importance inthe sequel. Let n ≥ 1 be an integer and Rn the standard Euclidean spaceof dimension n. It turns out that the special orthogonal (compact) group ofRn is of the form G(R) with G reductive over Z if, and only if, the integer nis congruent to −1, 0 or +1 modulo 8. Let us describe such a G under theassumption n ≡ 0 mod 8. It is well-known that in this case Rn possesses evenunimodular lattices. Such a lattice L is naturally equipped with an integralquadratic form, positive definite and non-degenerate over Z. The associateorthogonal group (scheme) OL is smooth over Z, and its neutral componentSOL is semisimple over Z, with real points SO(Rn).

We will denote by Ln the set of even unimodular lattices in Rn. Any twoelements of Ln are in the same genus, that is are isometric over Zp for eachprime p (hence over the rationals as well, according to Hasse and Minkowski).This implies first that the space A2(SOL) depends in an inessential way onthe choice of the lattice L. In order to fix ideas, we will focus in this memoiron the group SOn := SOEn , where En denotes the standard even unimodularlattice generated by 1

2(1, . . . , 1) and the n-tuples of integers (x1, . . . , xn) with∑

i xi is even. Another consequence is that we have a natural identification

Ln∼→ SOn(Q)\SOn(A)/SOn(Z)

compatible with the obvious actions of SOn(R) on both sides. If 1 denotesthe trivial representation of G(R), and if Xn = SO(Rn)\Ln denotes thefinite set of proper isometry classes of elements in Ln, we have thus natural

3This is an Archimedean analogue of Ramanujan’s conjecture, still open for generalcongruence subgroups.

v

isomorphisms

A1(SOn) ' {f : Xn → C} '⊕

{π ∈ Πdisc(SOn) |π∞'1}

m(π) πf .

The vector space C[Xn], dual of A1(SOn), is therefore an H(SOn)-module ina natural way. For instance, it is an exercise to see that the endomorphismof C[Xn], mapping the class of a lattice to the sum of the classes of its p-neighbors, is induced by an element of Hp(SOn) that we will denote by Tp.The determination of this endomorphism is exactly the problem considered atthe beginning of the introduction.4 Let us add that the spaces AU(SOn), withU arbitrary (but necessarily finite dimensional), have similar interpretationsas spaces of SOn(R)-equivariant functions Ln → U∗; many such spaces willplay a role in this memoir.

Langlands’ functoriality principle

Let us describe in a rather brief way Langlands’ general conjectures in thecase of level 1 automorphic forms. A starting point is the notion, introducedby Langlands, of dual group. If G is reductive over Z, its dual in the sense ofLanglands is simply “the” complex linear algebraic reductive group, denotedG, whose based root datum is dual (or inverse) to that of GC:

GC GLn PGLn Sp2g PGSp2g SO2n+1 SO2n PGSO2n

G GLn SLn SO2g+1 Spin2g+1 Sp2n SO2n Spin2n

This group allows first Langlands to parameterize the elements of Π(G).He observes that the Satake isomorphism provides a canonical bijection, foreach prime p, between the set of ring homomorphisms Hp(G) → C and theset of semisimple conjugacy classes in G(C). In a similar way, he interpretsthe infinitesimal character (in the sense of Harish-Chandra) of a unitaryrepresentation of G(R) as a semisimple conjugacy class in the Lie algebraof G. In the end, to each element π of Π(G) is associated a collection ofconjugacy classes

c(π) = (c∞(π), c2(π), c3(π), · · · )

which uniquely determines πp for each prime p premier, as well as the in-finitesimal character of π∞, which only leaves finitely many possibilities for

4Actually, we will mostly consider the analogous problem, only slightly simpler, inwhich SOn is replaced by On := OEn

, whose only flaw is that it does not quite fit theconventions adopted here, as On is not connected, but this slight difference is inessential.

vi

π∞. These parameterizations, recalled in chapter VI, have some very con-crete aspects. For instance, we will see that for π in Π(SOn) we have therelation:

(2) πp(Tp) = pn2−1 trace cp(π).

Let G and G′ be two reductive groups over Z, and a morphism of algebraicgroups r : G → G′. Langlands’ functoriality principle predicts, for eachconstituent π of A2(G), the existence of a constituent π′ of A2(G′) whichcorresponds to π, in the sense that we have an equality of conjugacy classesr(cv(π)) = cv(π

′) for each v in the set V := {∞, 2, 3, 5, . . . } of all the places ofQ. It is only a principle, rather than a conjecture, as it is not quite accurate asstated, even after a reasonable sense being given to the term “constituent". Inwhat follows, we propose ourselves to make the statement of the functorialityprinciple precise in the important case G′ = GLn, in which r is nothing elsethan an n-dimensional representation of the algebraic group G. We will laterrefer to this statement as the Arthur-Langlands conjecture.

The Langlands group of Z

Langlands has observed that the formulation of his conjectures is enlight-ened if one assumes the existence of a certain group, that we will denotehere by5 LZ, whose representations in G parameterize the automorphic rep-resentations of G in an appropriate sense. We may think of this group asbeing an extension of the absolute Galois group of Z (... trivial according toMinkovski!). For our needs in this preface, we will only assume that LZ is acompact Hausdorff topological group (hence an inverse limit of compact Liegroups) satisfying the axioms denoted (L1), (L2) and (L3) that we are goingto introduce below.

(L1) LZ is equipped, for each prime p, with a conjugacy class Frobp ⊂ LZ,and its complex pro-Lie-algebra, with a semisimple conjugacy class Frob∞.

Let G be reductive over Z. Following Arthur and Langlands, we denote byΨ(G) the set of G(C)-conjugacy classes of continuous group homomorphisms

(3) ψ : LZ × SL2(C) −→ G(C)

5To be completely honest, Langlands rather considers a group which applies to allautomorphic forms, rather that to level 1 forms only, of which our LZ would merely bea quotient [Lan79, §2]. Moreover, following Arthur in [Art89, §8], we adopt Kottwitz’spoint of view [Kot84, §12] on the Langlands group, which amounts to see it as a topologicalgroup rather than a pro-algebraic one as Langlands does.

vii

which are polynomial on the SL2(C)-factor. Such a ψ is called discrete if thecentralizer Cψ of Im ψ in G(C) is finite modulo the center Z(G) of G(C).For example, if G is GLn, in which case we also have G = GLn and ψ isnothing else than an n-dimensional representation of LZ × SL2(C), then ψis discrete if, and only if, it is an irreducible representation. We denote byΨdisc(G) ⊂ Ψ(G) the subset of classes of discrete morphisms.

In parallel with what has been done for Π(G), Arthur and Langlands as-sociate to each ψ in Ψ(G) a collection of conjugacy classes c(ψ) = (cv(ψ))v∈V

defined by c∞(ψ) = ψ(Frob∞, e∞) and cp(ψ) = ψ(Frobp, ep), where the evare respectively the elements of sl2(C) for v = ∞, and of SL2(C) for v = p,defined by

e∞ =

[−1

2 00 1

2

]and ep =

[p−

12 0

0 p12

].

(L2) For each integer n ≥ 1, there is a unique bijection

π 7→ ψπ, Πdisc(GLn)∼→ Ψdisc(GLn),

such that we have c(π) = c(ψπ) for all π ∈ Πdisc(GLn). Moreover, ψπ istrivial on SL2(C) if, and only if, we have π ∈ Πcusp(GLn).

This axiom, together with the compactness of LZ, contains the generalizedRamanujan conjecture. It also shows |Lab

Z | = dimA(GL1) = 1. We will provein Chapter IX that (L2) implies as well that LZ is connected.

(L3) For each G reductive over Z, there exists a decomposition

Adisc(G) =⊥⊕

ψ∈Ψdisc(G)Aψ(G),

stable under G(R) and H(G), and satisfying the following property: if π ∈Π(G) appears in Aψ(G) then we have c(π) = c(ψ).

It is Arthur’s idea that the failure of Ramanujan’s conjecture may beentirely explained in general by the presence of SL2(C) in the definition ofΨ(G) (Formula (3)).

Arthur and Langlands strengthen the axiom (L3) by adding a conversestatement, called the multiplicity formula, whose formulation requires how-ever the introduction of more technical ingredients. Let us simply say thatif ψ ∈ Ψdisc(G) and π ∈ Π(G) satisfy c(π) = c(ψ), this formula expresses

viii

the multiplicity of π in the subspace Aψ(G) as the scalar product of two“explicit”6 characters of the finite group Cψ/Z(G).

Arthur-Langlands’ conjecture

Let us go back to the statement of the Arthur-Langlands conjecture al-luded to above. We assume first the existence of a compact group LZ satisfy-ing the axioms (L1), (L2) and (L3). Let G be reductive over Z, π in Πdisc(G)

and r : G→ GLn a representation. Let ψ ∈ Ψdisc(G) be such that π appearsin Aψ(G), such a ψ exists by (L3). The decomposition into irreducibles ofthe representation r ◦ ψ of the direct product LZ × SL2(C) may be writtenas ⊕i ri ⊗ Symdi−1C2 for some irreducible representation ri of dimension niof LZ, and some integers di ≥ 1. According to (L2), we have ri ' ψπi for aunique πi in Πcusp(GLni

). In particular, for each v ∈ V we have the identitybetween conjugacy classes

(4) r(cv(π)) =⊕i

cv(πi)⊗ Symdi−1(ev)

(the reader will have no trouble decrypting the meaning of the right-handside of this equality).

As a consequence, if we have π in Πdisc(G) and a representation r : G →GLn, one conjectures the existence of a decomposition n =

∑i nidi and of

representations πi in Πcusp(GLni), unique up to a permutation of the indices

i, satisfying equality (4): this is the precise form of the Arthur-Langlandsconjecture that had been promised. Observe that LZ does not appear at allin this formulation.

In his work that we already mentionned, Arthur has proved the followingspecial cases of this conjecture: GQ is either the symplectic group Sp2g ofa symplectic space sur Q of dimension 2g, or the special orthogonal groupof a quadratic space of dimension 2n (resp. 2n + 1) over Q possessing atotally isotropic subspace of dimension n, π ∈ Πdisc(G) is arbitrary, and r

is the natural representation of G, called the standard representation, whoserespective dimension is 2g + 1, 2n and 2n. For such groups, Arthur also

6The definition of these characters is very delicate. One of them is a group homomor-phism Cψ/Z(G)→ C× defined by Arthur in [Art89, p. 55] with the help of the ε-factorsof certain L-functions associated to ψ. The other one depends on the definition of a cer-tain finite subset of irreducible unitary representations of G(R) associated to ψ, denotedΠ∞(ψ), now usually called an Arthur packet [Art89, §4]. This character is nonzero if, andonly if, π∞ belongs to Π∞(ψ). In the important special case Cψ = Z(G), the multiplicityof π in Aψ(G) is thus nonzero if, and only if, we have π∞ ∈ Π∞(ψ).

ix

proves a version of the multiplicity formula to which we alluded during thediscussion of the axiom (L3). We shall state more precise forms of Arthur’sresults in chapter VIII of the memoir. Let us stress however that we shallnot say anything about Arthur’s proofs: they go far beyond the scope of thiswork.

Galois representations and motives

The group LZ is subject to several other conjectures. A most temptingone is that it satisfies the Sato-Tate property: the Frobp are equidistributedin the set of conjugacy classes of LZ, equipped with its invariant probabilitymeasure.7 We will rather discuss in this paragraph the conjectural relationbetween LZ, Grothendieck motives and Galois representations.

These links will only concern the quotient of LZ whose irreducible rep-resentations parameterize, in the sense of axiom (L2), the representationsπ in Πcusp(GLn) which are algebraic. This adjective means here that, if wedenote by λi the eigenvalues of the conjugacy class c∞(π) ⊂ Mn(C), wehave λi − λj ∈ Z for all i, j. We then denote by w(π) the maximum of thedifferences λi − λj, and call it the motivic weight of π.

Denote by Q ⊂ C the subfield of algebraic numbers. Fix a prime `, Q`

an algebraic closure of the field of `-adic numbers, and ι : Q → Q` an em-bedding. Thanks to the works of a number of mathematicians (includingClozel, Deligne, Fontaine, Grothendieck, Langlands, Mazur, Serre, Shimura,Taniyama, Tate, Weil ...) one conjectures the existence of a natural bijec-tion π 7→ ρπ,ι between the set of algebraic π in Πcusp(GLn), and the set ofisomorphism classes of irreducible continuous representations Gal(Q/Q) −→GLn(Q`) which are unramified at each prime p 6= ` and crystalline at ` inthe sense of Fontaine, with smallest Hodge-Tate weight 0. One asks that thisbijection satisfies in particular the equality 8

det(t− ρπ,ι(Frobp)) = ι(det(t− pw(π)/2cp(π)))

for each prime p 6= `, which determines it uniquely.

This conjecture may be readily seen as an “algebraic” analogue of the ax-iom (L2). A number of difficult and important special cases of it are known.According to Fontaine and Mazur, one expects that the Galois representa-tions above are exactly those appearing in the `-adic realizations of puremotives over Q with everywhere good reduction.

7Given the connectedness of LZ, it would be easy to see that this property impliesindeed the usual Sato-Tate conjecture for modular forms for SL2(Z).

8This equality makes sense as we also conjecture that we have det(t− cp(π)) ∈ Q[t] ifπ is algebraic.

x

Conclusion

Let G be reductive over Z and r a representation of G. As we have seen,the Arthur-Langlands conjecture predicts that for each π in Πdisc(G) the col-lection of conjugacy classes r(c(π)) may be expressed in a very precise wayin terms of building blocks which are some elements πi of Πcusp(GLni

), andsome integers di, with dim r =

∑i nidi. Here are some questions which nat-

urally arise: assume that a representation π in Πdisc(G) is given, for instancesuch that πf appears concretely in a specific AU(G), can we determine theassociated representations πi and integers di? is it easier to determine themrather than π itself ?

A first obstacle that we encounter, if one tries to illustrate these questions,is to have at our disposal examples of groups G and of irreducible unitaryrepresentations U of G(R) for which we know how to determine whetherAU(G) is nonzero, or better its dimension. When U is a discrete seriesrepresentation, this is an accessible but notoriously difficult problem: whenG = Sp2g it contains for instance the question of determining the dimensionof spaces of Siegel modular cuspforms for Sp2g(Z), a classical problem thathas been solved only very recently by Taïbi in genus 3 ≤ g ≤ 7. WhenU is not in the discrete series, it seems hopeless to obtain a formula fordimAU(G), as is shown by the example G = PGL2.

The special case G(R) compact, for which all the irreducible representa-tions are in the discrete series, has the peculiar feature that the question ofdetermining dimAU(G) is significantly more elementary. We will give manysuch examples with G = SOn. The case G = SO24 is especially interestingfrom this point of view, as it is one of the groups of highest rank for whichdimAU(G) can be computed for at least one U (and with AU(G) 6= 0). In-deed, we have dimA1(G) = |X24| and this cardinality is 25 as the Leechlattice is the only one, among the 24 Niemeier lattices, not to admit anyimproper isometry. We are forced to ask ourselves the following question:

Question 0. Let r be the standard representation of SO24 and π in Πdisc(SO24)with π∞ = 1; can we determine the collection of representations πi and theintegers di corresponding to π and r according to Arthur-Langlands’ conjec-ture ?9

This is the question at the origin of this work. Formulas (4) and (2) show9Observe that Arthur’s results do not immediately apply here as SOn is not (quasi-)

split over Q. Nevertheless, we will prove that the Arthur-Langlands conjecture is satisfiedwhen π and r are as in the statement of Question 0, by applying Arthur’s results to Sp2g

and using some theta series arguments.

xi

that a positive answer to this question gives decisive informations about thep-neighbor problem in dimension 24.

Before saying more about Question 0, let us add that the πi which appearin its statement are not arbitrary: they are algebraic. More generally, if G isreductive over Z and if π is in Πdisc(G) with π∞ a discrete series representa-tion, then the eigenvalues of c∞(π) in the adjoint representation of Lie G arein Z (Harish-Chandra); it follows that if r is an arbitrary representation of Gthen the representations πi associated to π and r by the Arthur-Langlandsconjecture are necessarily algebraic. As a consequence, those π are related tomotives and Galois representations, which makes them even more interesting.Those links are deep. We will show for instance that Arthur’s multiplicityformula suggests that if π in Πcusp(GL8k) is algebraic, isomorphic to its dual,and if the eigenvalues of c(π∞) are distinct, then there exists π′ in Πdisc(SO8k)satisfying r(c(π′)) = c(π). These unexpected relations between Galois rep-resentations and even unimodular lattices clearly show the interest to studyΠdisc(SOn) for the arithmetician.

We now go back to Question 0. An obstacle that we faced immediately, atleast when we started working on this question, is that very few results wereknown about Πcusp(GLn) with n > 2, even if we restrict ourselves to algebraicrepresentations. 10 For instance, assuming there was a representation π inΠdisc(SO24) satisfying π∞ = 1, and such that one of the associated πi was inΠcusp(GLni

) with ni big, then it is very likely that we would never be ableto say anything interesting neither about this π, nor about the p-neighborproblem in dimension 24. Observe that we always have ni ≤ 24, but alsow(πi) ≤ 24, by an inspection of c∞(π).

One of our main results will be the proof of a classification of the auto-morphic representations π in Πcusp(GLn), with n ≥ 1 arbitrary, which arealgebraic of motivic weight w(π) ≤ 22. We will see that there are only 11such representations, and that they all appear (as πi) in the answer to Ques-tion 0. We have furthermore n ≤ 4 in all cases, with exactly four of themin Πcusp(GL4). These four representations, which actually come from certainvector valued Siegel cuspform of genus 2, will play an important role in thismemoir.

The scope of the classification above is broader: G being arbitrary, the10The situation is very different by now, thanks to the works [CR12] and [Tai14]. Note

that although these works have been published before the present memoir, they haveactually been entirely motivated by it. A lot of important questions remain, for examplewe do not know the number of algebraic π in Πcusp(GL3) with a given Archimedeancomponent π∞.

xii

Arthur-Langlands conjecture suggests that any representation π in Πdisc(G)with π∞ is the discrete series, and such that c∞(π) is “small enough”, is builtfrom the 11 above automorphic representations. For example, we will seehow to use this approach to determine the dimension of the space of Siegelmodular cuspforms of weight ≤ 12 for Sp2g(Z), for any genus g ≥ 1.

It seems reasonable to end this preface here, and to leave the reader thepleasure to immerse themselves in the actual introduction of the memoir.

** *

The author warmly thank Colette Moeglin, David Renard and OlivierTaïbi for useful discussions, as well as Richard Borcherds and Thomas Mé-garbané for their remarks.

xiii

I. Introduction

1. Even unimodular lattices

Let n ≥ 1 be an integer and consider the Euclidean space Rn equipped withits standard inner product (xi) · (yi) =

∑i xiyi. An even unimodular lattice

of rank n is a lattice L ⊂ Rn of covolume 1 such that x · x is an even integerfor all x in L. The set Ln of those lattices is equipped with an action of theEuclidean orthogonal group O(Rn), and we denote by

Xn := O(Rn)\Ln

the set of isometry classes of even unimodular lattices of rank n. To each Lin Ln, one can attach a quadratic form

qL : L→ Z, x 7→ x · x2,

whose associated bilinear form x · y has determinant 1. The map L 7→ qLinduces a bijection between Xn and the set of isomorphism classes of positivedefinite quadratic forms of rank n over Z whose determinant is 1.

As is well known, the set Xn is finite. It is not empty if and only if wehave n ≡ 0 mod 8. A standard example of an element of Ln is the lattice

En := Dn + Ze

where Dn = {(xi) ∈ Zn,∑

i xi ≡ 0 mod 2}, e = 12(1, 1, . . . , 1), and n ≡

0 mod 8. Let us explain this notation. To each element L of Ln is associateda root system (of type ADE)

R(L) := {x ∈ L, x · x = 2},

whose rank is ≤ n. The root system R(E8) is then of type E8 and generatesover Z the lattice E8. For n > 8, R(En) is of type Dn and generates Dn.

The set Xn has only been determined so far in dimension n ≤ 24. Mordelland Witt have respectively proved

X8 = {E8} and X16 = {E16,E8 ⊕ E8}.

1

The two lattices E16 and E8 ⊕ E8 will play an important role in this mem-oir. They are at the same time easy and hard to distinguish : their rootsystems are different, but they represent each integer exactly the same num-ber of times. This last and rather famous property leads for instance to theisospectral tori discovered by Milnor.

The elements of X24 have been classified by Niemeier [Ni73], who proved inparticular |X24| = 24. Before saying more about those lattices, let us mentionthat for n ≥ 32 the Minkowski-Siegel-Smith mass formula shows that the sizeof Xn explodes. As an example, we have |X32| > 8 ·106 [Ser70]; there is evenmore than a billion of elements in X32 according to King [Kin03].

An element of L24 is called a Niemeier lattice. The most famous Niemeierlattice is the Leech lattice. Up to isometry, it is the unique element L inL24 such that R(L) = ∅ (Conway). If L is a Niemeier lattice which is notisomorphic to the Leech lattice, a remarkable fact is that its root systemR(L) has rank 24 and all its irreducible components share the same Coxeternumber. A simple proof of this fact has been given by Venkov [Ven80].The miracle is then that the map L 7→ R(L) induces a bijection betweenX24 − {Leech} and the set of isomorphism classes of root systems R of rank24 whose irreducible components are of type ADE and share the same Coxeternumber h(R). The proof is a rather tedious case-by-case verification.

R D24 D16E8 3E8 A24 2D12 A17 E7 D10 2E7 A15 D9

h(R) 46 30 30 25 22 18 18 16

R 3D8 2A12 A11 D7 E6 4E6 2A9D6 4D6 3A8 2A7 2D5

h(R) 14 13 12 12 10 10 9 8

R 4A6 4A5 D4 6D4 6A4 8A3 12A2 24A1

h(R) 7 6 6 5 4 3 2

Table I.1: The 23 equi-Coxeter root systems of type ADE and rank 24

The results mentioned in this paragraph are explained in Chapter II, whosemain aim is to recall some classical results. We first develop some prereq-uisites of bilinear and quadratic algebra. They are necessary to understandthe constructions of quadratic forms alluded above, as well as others that weshall need throughout this memoir. In particular, we recall Venkov’s theoryand explain the construction of certain Niemeier lattices. We also take theopportunity to recall some basic facts about the classical group schemes overZ that we shall use in the sequel. Appendix B contains in particular a vari-ant of the results of Chapter II: we study the even Euclidean lattices with

2

determinant 2 as well as the corresponding theory of quadratic forms over Z(in odd dimensions).

2. Kneser neighbors

Let p be a prime. The notion of p-neighbors has been introduced by M.Kneser. It can be viewed as a tool to construct a collection of even unimod-ular lattices from a given one and the prime p. In Chapter III, we studyseveral variations of this notion and give many examples.

Following Kneser, we say that two lattices L,M in Ln are p-neighbors ifL ∩ M has index p in L (hence in M). It is easy to construct all the p-neighbors of a given even unimodular lattice L. Indeed, to each isotropic line` in L⊗Fp, say generated by an element x in L such that qL(x) ≡ 0 mod p2,let us associate the even unimodular lattice1

voisp(L; `) := H + Zx

p,

where H = {y ∈ L, x · y ≡ 0 mod p} (this lattice does not depend on thechoice of x). The map ` 7→ voisp(L; `) induces a bijection between CL(Fp) andthe set of p-neighbors of L, where CL denotes the projective (and smooth)quadric over Z defined by qL = 0. This quadric turns out to be hyperbolicover Fp for each prime p, thus the number of p-neighbors of L is exactly|CL(Fp)| = 1 + p+ p2 + · · ·+ pn−2 + p

n2−1, a number that we shall also denote

by cn(p).

Consider for instance the element ρ = (0, 1, 2, . . . , 23) of E24. It generatesan isotropic line in E24⊗F47 because of the congruence

∑23i=1 i

2 ≡ 0 mod 47.It is not very difficult to check that vois47(E24; ρ) has no root; we thus havean isometry

vois47(E24; ρ) ' Leech.

This especially simple construction of the Leech lattice is attributed to Thomp-son in [CS99], we shall come back to it later. It illustrates the slogan claimingthat many constructions of lattices are special cases of constructions of neigh-bors.

Going back to the general setting, we obtain for each L in Ln a partitionof the finite quadric CL(Fp) given by the isometry class of the associatedp-neighbor of L. One of the aims of this memoir is to study this partitionin dimension n ≤ 24. For instance, can we determine the number Np(L,M)

1The notation vois comes from the french word voisin for neighbor .

3

of p-neighbors of L which are isometric to another given M in Ln ? Thefirst interesting case of this question occurs of course in dimension n = 16.In order to formulate the result, it will be convenient to introduce the linearmap Tp : Z[Xn] → Z[Xn] defined by Tp [L] =

∑[M ], the sum being over all

the p-neighbors M of L.

Theorem A. In the basis E8 ⊕ E8, E16 the matrix of Tp is

c16(p)

[1 00 1

]+ (1 + p+ p2 + p3)

1 + p11 − τ(p)

691

[−405 286405 −286

],

where τ is Ramanujan’s function, defined by q∏m≥1

(1− qm)24 =∑

n≥1 τ(n)qn.

For instance, this statement asserts that for each prime p we have theformula Np(E8 ⊕ E8, E16) = 405

691(1 + p11 − τ(p))p

4−1p−1

. Although Theorem Awas probably known to specialists, we have not found it stated this way in theliterature. Throughout this memoir, we shall give several proofs of it. Giventhe theory of theta series and modular forms for SL2(Z), the presence of τ(n)in the statement may look quite classical at first sight. For instance, if we setrL(n) = |{x ∈ L, x·x = 2n}|, it is easy to show rLeech(p) = 65520

691(1+p11−τ(p))

for each prime p, a formula that looks similar to the one of the theorem.Nevertheless, the presence of the term τ(p)p

4−1p−1

in the counting problemabove appears to be quite subtler; it will turn out to be equivalent to a nontrivial case of the Arthur-Langlands functoriality conjecture.2

Our main theorem is a statement similar to the one of Theorem A butconcerning the Niemeier lattices. It would be possible to give a formulation ofit in the same style as the one of this theorem, namely an explicit formula for amatrix of Tp on Z[X24], although it would look very indigestible. This explicitformula actually involves rational coefficients with such big denominatorsthat it appears quite remarkable from their inspection that Np(L,M) maybe an integer ! We will state a conceptual (and equivalent) version of ourresult in §I.4 below (Theorem E). A remarkable feature is that all the cuspidalmodular forms of weight k ≤ 22 for the group SL2(Z), as well as 4 vector-valued Siegel modular forms for Sp4(Z), appear in the statement. Let usdiscuss from now on some consequences concerning the Niemeier latticesthat will follow in fine from an analysis of our formulas.

2Incidentally, the comparison between theorem A and the formula above for rLeech(p)

leads to the “purely quadratic” relation Np(E8 ⊕ E8,E16) = 91456 · rLeech(p) · p

4−1p−1 , that

we don’t know how to prove directly.

4

Consider the graph Kn(p) whose set of vertices is Xn, and with an edgebetween two different classes L,M in X24 if and only if we have Np(L,M) 6= 0.Kneser has proved that Kn(p) is connected, for all n and p, as a consequenceof his celebrated strong approximation theorem. This nice result implies thatwe can theoretically reconstruct Xn from the single lattice En and a primep. This was actually useful to Niemeier in his computation of X24 using2-neighbors.

The graph K16(p) is the connected graph with 2 vertices, thanks to Kneser.This is of course compatible with the estimate |τ(p)| < 2p

112 (Deligne-Rama-

nujan) and with the formula for Np(E8⊕E8,E16) given by Theorem A. On theother hand, the graph K24(2), determined by Borcherds [CS99], is not trivialat all. It has diameter 5 and this Wikipedia page http://en.wikipedia.org/wiki/Niemeier_lattice gives a lovely representation of it, also due toBorcherds. Our results allow for instance to determine the graph K24(p) foreach prime p (§X.2). 3

Theorem B. (i) Let L be a Niemier lattice with roots. Then L is a p-neighbor of the Leech lattice if and only if p ≥ h(R(L)).

(ii) The graph K24(p) is complete if and only if p ≥ 47.

Let us make some remarks about this statement. The first point concernsthe constructions of the Leech lattice as a p-neighbor of a Niemeier latticewith roots. For instance, we observe on the Borcherds graph K24(2) thatLeech is at distance 5 from E24, and that it is only linked with the latticewith root system 24A1 (which is the Niemeier lattice with roots whose con-struction is the most delicate, as it uses the Golay code, §II.3). This lastproperty is quite simple to understand: if the Leech lattice is a 2-neighbor ofa Niemeier lattice L (with roots), then L has an index 2 subgroup withoutroots. In particular, R(L) has the property that the sum of two roots is nota root, so that its irreducible constituents must have rank 1, which forcesR(L) = 24A1. Among the root systems of table I.1, this root system is theunique one whose Coxeter number is 2, which fits assertion (i) of Theorem B.

The most elementary part of Theorem B, proved in §III.4 and generaliz-ing the previous observations, amounts to show the inequality p ≥ h(R(L)) ifLeech is a p-neighbor of L. This is an analog of a result of Kostant [Kos59]asserting that the minimal order of a finite order regular element in a con-nected, compact, adjoint Lie group, coincides with the Coxeter number of its

3A list of those graphs is available at the address http://gaetan.chenevier.perso.math.cnrs.fr/niemeier/niemeier.html.

5

root system. On the other hand, the proof of the other assertions requiresthe use of Theorem E as well as a number of Ramanujan style inequalities;it will be completed only in Chapter X (§X.2 and §X.3).

In Chapter III, we also study the limit cases of the assertion (i) of TheoremB (the arguments are direct, i.e. they do not rely on Theorem E). Forthat purpose, we proceed to a detailed analysis of the elements c of CL(Fp)satisfying voisp(L; c) ' Leech, where L is a Niemeier lattice with non-emptyroot system R = R(L). It is necessary, for the pertinence of the statements,to study more generally the d-neighbors of L, where d ≥ 1 is a non-necessarilyprime integer (§III.1). We show that if ρ is a Weyl vector of R, and if we seth = h(R), then we have isometries (Theorem III.4.2.10)

(2.1) voish(L; ρ) ' voish+1(L; ρ) ' Leech.

This makes sense since we have ρ ∈ L (Borcherds) and qL(ρ) = h(h + 1)(Venkov). This statement contains for instance the aforementioned obser-vation of Thompson. Actually, these 23 (or 46) constructions of the Leechlattice are nothing else than the famous holy constructions due to Conwayand Sloane [CS82]. However, we give a new proof of the isometries (2.1)using the theory of neighbors, and we show as well the identities

(2.2) Nh(L,Leech) =|W |ϕ(h)g

and Nh+1(L,Leech) =|W |h+ 1

,

where W denotes the Weyl group of R, and g2 is the index of connexion of Rin the sense of Bourbaki. We conclude Chapter III by a study of vois2(L; ρ)inspired by results of Borcherds (see Figure III.1).

3. Theta series and Siegel modular forms

Let us come back to the determination of the operator Tp on Z[Xn]. Westart with some simple observations: the Tp commute each others and areself-adjoint for a suitable inner product on R[Xn] [NV01] (§III.2). We havethus to find a basis of common eigenvectors of the Tp, as well as the associatedcollections of eigenvalues. There is only one obvious stable line, generatedby∑

L∈Xn

[L]|O(L)| , on which the operator Tp has the “trivial” eigenvalue cn(p).

As already seen in the preface, we are actually in the presence of a dis-guised problem belonging to the spectral theory of automorphic forms. In-deed, if G = On denotes the orthogonal group scheme over Z defined bythe quadratic form qEn , and if A denotes the adèle ring of Q, genus the-ory leads to an isomorphism of G(R)-sets Ln = G(Q)\G(A)/G(Z) (§II.2,

6

§IV.1). It follows that the dual of R[Xn] is naturally isomorphic to the spaceof real-valued functions on G(Q)\G(A) which are invariant under the actionof G(R)×G(Z) by right translations. In this description, the operator Tp isinduced by a specific element of the ring H(G) of Hecke operators of G.

Those classical observations are recalled in Chapter IV. Although we shallbe mostly interested in the automorphic forms of the Z-group On, our state-ments and proofs will require the introduction of several variants of them (au-tomorphic forms for SOn, PGOn and PGSOn), modular forms for SL2(Z),vector-valued Siegel modular forms for Sp2g(Z), and even, by the use ofArthur’s results, some automorphic forms for PGLn. It will thus be necessaryto adopt from the beginning a sufficiently general point of view embracing allthese objects (§IV.3). The reader will find in §IV.1 and §IV.2 an elementaryexposition of Hecke operators. The emphasis is placed on the examples givenby the classical groups and their variants (Hecke, Satake, Shimura); they leadto a deeper point of view on p-neighbors and their generalizations. In §IV.4and §IV.5, we review some properties of automorphic forms of On and ofSiegel modular forms. Let us emphasize that the writing in this chapter isintended for non-specialists, and pretends to little originality.

An approach to study the H(On)-module Z[Xn] is to examine the Siegeltheta series ϑg(L) of each genus g ≥ 1 of the elements L of Ln. For alln ≡ 0 mod 8 and all g ≥ 1, they allow us to define a linear map

ϑg : C[Xn]→ Mn2(Sp2g(Z)), [L] 7→ ϑg(L),

where Mk(Sp2g(Z)) denotes the space of Siegel modular forms of weight k ∈ Zfor Sp2g(Z) (§V.1). The relevance of this map for our problem comes from the(generalized) Eichler commutation relations; they assert that ϑg intertwineseach element of H(On) with a certain “explicit” element of H(Sp2g) (Eichler,Freitag, Yoshida, Andrianov, §V.1). The map ϑg is trivially injective forg ≥ n. It seems however quite difficult in general to determine the structureof the H(Sp2g)-module Mk(Sp2g(Z)), especially when g grows. Nevertheless,we will develop in Chapter IX a strategy allowing to solve some new cases ofthis problem, that uses among other things recent results of Arthur [Art13].

The map ϑg has been studied by several authors. Its kernel, which de-creases when g grows, describes the linear relations between the genus g thetaseries of the elements of Ln, and the description of its image is an instance ofEichler’s famous basis problem. More precisely, ϑg induces an injective map

(3.1) Kerϑg−1/Kerϑg −→ Sn2(Sp2g(Z))

where Sk(Sp2g(Z)) ⊂ Mk(Sp2g(Z)) denotes the subspace of cuspforms (see§V.1 or the footnote below for the convention on ϑ0), and Eichler asks

7

whether this map is surjective. An important result of Böcherer [Boc89]gives a necessary and sufficient condition for an eigenform for H(Sp2g) to bein its image, in terms of the vanishing at a certain integer of an associatedL-function (§VII.2).

The case n = 16.

The case n = 16 is the subject of a famous story, recalled in §V.2. Indeed,a classical result of Witt and Igusa asserts that we have

(3.2) ϑg(E8 ⊕ E8) = ϑg(E16) if g ≤ 3.

These remarkable identities mean that E8⊕E8 and E16 represent each positiveintegral quadratic forms of rank ≤ 3 exactly the same number of times. Thisis well known in genus g = 1, as a consequence of the vanishing S8(SL2(Z)) =0 (and leads to the aforementioned Milnor’s isospectral tori). This showsincidentally4 that “the” non-trivial eigenvector in Z[X16] is [E16]− [E8 ⊕ E8].The difficulty in genera 2 and 3 is that although the vanishing of S8(Sp2g(Z))still holds, it is more complicated to prove. In Appendix A, we present adifferent and ingenious proof of the identities (3.2) due to Kneser, whichdoes not rely on any such vanishing results.

The Siegel modular form J = ϑ4(E8 ⊕ E8) − ϑ4(E16), which is nothingelse than the famous Schottky form, is easily shown to be non-zero. Fol-lowing Poor and Yuen [PY96], we even know that it generates S8(Sp8(Z)).Theorem A follows then from the resolution by Ikeda [Ike01] of the Duke-Imamoğlu conjecture [BK00]. Indeed, applied to Jacobi’s modular form ∆in S12(SL2(Z)), Ikeda’s theorem shows the existence of a non-zero Siegelmodular form in S8(Sp8(Z)), which is an eigenform for H(Sp8), whose Heckeeigenvalues are explicitly determined by the τ(p). Ikeda’s proof is quite dif-ficult, and our main contribution to Theorem A in this memoir is to havefound a very different proof of Ikeda’s result in this specific case.

The important result is the following. For any map f : Ln → C, we definea map Tp(f) : Ln → C by setting, for any L ∈ Ln, Tp(f)(L) =

∑M f(M),

the sum being over all the p-neighbors M of L. If 1 ≤ g ≤ n/2, we denoteby Hd,g(Rn) the space of polynomials (Rn)g → C which are harmonic for theEuclidean Laplace operator on (Rn)g, and which satisfy P ◦γ = (det γ)d P forall γ ∈ GLg(C) (§V.4). This space is equipped with a linear representationof O(Rn) in a natural way.

4This assertion can be proved in a much more direct manner. Indeed, if ϑ0 denotes thelinear map C[Xn] → C sending to 1 the class of any element of Ln, the obvious equalityϑ0 ◦ Tp = cn(p)ϑ0 shows that Kerϑ0 is stable by Tp.

8

Theorem C. Let q +∑

n≥2 anqn be a modular form of weight k for SL2(Z)

which is an eigenform for the Hecke operators, and let d = k2− 2. There

exists a map f : L8 → C such that :

(i) for each prime p, we have Tp(f) = p−d p4−1p−1

ap f ,

(ii) f generates under O(R8) a representation isomorphic to Hd,4(R8).

The paragraph V.4 is mostly devoted to the proof of the special casek = 12 of this theorem. It leads to a complete and rather elementary proofof Theorem A. The general case will be addressed, and made more precise,in §VII.2.

Let us give an idea of the proof. The first step is to realize the mod-ular form we are starting from as a theta series

∑x∈E8

P (x) qx·x2 , where

P : R8 → C is a suitable harmonic polynomial. In the case of the modularform ∆, we check that any non-zero harmonic polynomial of degree 8 whichis invariant under the Weyl group W(E8) does the trick, and in general we in-voke a result of Waldspurger [Wal79]. This construction defines a subspaceof functions L8 → C which are eigenvectors for the Hecke operators in H(O8),with eigenvalues related to the ap by the Eichler commutation relations, andwhich generate under O(R8) a representation isomorphic to H8,1(R8). Themain idea is to apply to them, at the source, an automorphism of order 3of L8 arising from triality. Such an automorphism is constructed from astructure of Coxeter octonions on the E8 lattice, and from an isomorphismL8 ' G(Q)\G(A)/G(Z) where G = PGSO8. The resulting functions satisfythe theorem: we refer to §V.4 for the details.

Condition (ii) of the statement implies that the function f gives birth to aSiegel theta series of genus 4 (with “pluriharmonic” coefficients). When thisseries does not vanish, we observe that is a substitute for the Ikeda lift ofgenus 4 of the modular form we started from. As it is not difficult to checkthis non-vanishing when k = 12, Theorem A follows.

Let us mention that we shall prove later the vanishing of S8(Sp2g(Z)) forall g 6= 4 (Theorem IX.5.10). When g = 5, 6, this had already been obtainedby Poor and Yuen [PY07], by different methods. As a consequence, the mapϑg : C[X16]→ M8(Sp2g(Z)) is surjective for any genus g ≥ 1.

9

The case n = 24

This case has been the subject of remarkable works by Erokhin [Ero79],Borcherds-Freitag-Weissauer [BFW98] and Nebe-Venkov [NV01] (§V.3). Ero-khin has showed the vanishing Kerϑ12 = 0, and the three authors of [BFW98]have proved that Ker ϑ11 has dimension 1. Nebe and Venkov have studiedloc. cit. the whole filtration of Z[X24] given by the sequence of Kerϑg forg ≥ 1. Their starting point is an explicit computation of the operator T2 onZ[X24] that they deduce from results of Borcherds (§III.3.3). They observethat its eigenvalues are distinct integers, which allows them to exhibit anexplicit basis of Q[X24] made of common eigenvectors for all the Tp. Theypropose as well a conjecture for the dimension of the image of the map (3.1)for each integer 1 ≤ g ≤ 12, that they prove in many cases, but not all. Weestablish their conjecture and even show that Eichler’s basis problem has apositive answer in dimension n = 24 for each genus 1 ≤ g ≤ 23 (TheoremIX.5.2 and Corollary IX.5.7).

Theorem D. The map ϑg : C[X24] → M12(Sp2g(Z)) is surjective, and in-duces an isomorphism Kerϑg−1/Kerϑg

∼→ S12(Sp2g(Z)), for each integerg ≤ 23. The dimension of S12(Sp2g(Z)) for g 6= 24 is given by the table:

g 1 2 3 4 5 6 7 8 9 10 11 12 > 12

dim S12(Sp2g(Z)) 1 1 1 2 2 3 3 4 2 2 1 1 0

We will give an idea of the proof in §I.6, the most difficult point being thefirst assertion. As a consequence of the theorem, we obtain a complete de-scription of the filtration (Kerϑg)g≥1 of Z[X24]. Let us mention that Eichler’sbasis problem has a negative answer in dimension n = 32 and genus g = 14,as we observe in Corollary VII.3.5.

4. Automorphic forms of classical groups

Siegel modular forms, as well as automorphic forms for On, can be studiedfrom the perspective of the recent results of Arthur [Art13]. If only toformulate Arthur’s results, we are forced to recall the most basic featuresof Langlands point of view on the theory of automorphic forms [Lan70][Bor77]. They are gathered in Chapter VI. The outlines of this point of

10

view have already been discussed in the preface. We shall recall briefly heresome of its aspects.

Let G be a semisimple5 group scheme over Z. We denote by Πdisc(G)the set of topologically irreducible subrepresentations of the space of square-integrable functions on G(Q)\G(A)/G(Z), for the natural actions of G(R)and of the (commutative) ring H(G) of Hecke operators of G (§IV.3). TheSatake isomorphism asociates to each π ∈ Πdisc(G), and to each prime p,a semisimple conjugacy class cp(π) in G(C), where G denotes the complexsemisimple algebraic group which is dual to GC in the sense of Langlands(§VI.1, §VI.2). This enlightening point of view on the eigenvalues of Heckeoperators, due to Langlands, is made explicit in §VI.2.8 in the case of classicalgroups and of the Hecke operators of interest in this memoir, following Gross’sarticle [Gro98]. Similarly, we recall how the Harish-Chandra isomorphismallows to view the infinitesimal character of the Archimedean component π∞of π as a semisimple conjugacy class c∞(π) in the Lie algebra of G (§VI.3).

As already explained in the preface, a central and structuring conjecture,originally due to Langlands “in the tempered case” and extended by Arthur ingeneral [Art89], asserts that those collections of conjugacy classes can all beexpressed in terms of the similar data relative to the elements of Πdisc(PGLm)for all m ≥ 1. This conjecture is discussed in §VI.4.4. Let us give here analternative way to state it, in terms of L-functions, which is especially easy tostate. Fix π ∈ Πdisc(G) and r : G(C)→ SLn(C) an algebraic representation.Following Langlands, the Euler product

L(s, π, r) =∏p

det(1− p−s r(cp(π)))−1

converges for Re s big enough. When G is the Z-group PGLm and r is thetautological representation of G = SLm, we simply write L(s, π) = L(s, π, r).The Arthur-Langlands conjecture for the couple (π, r) predicts the exis-tence of an integer k ≥ 1, and for i = 1, . . . , k of a representation 6 πiin Πcusp(PGLni

) and an integer di ≥ 1, such that the following equality holds(§VI.4.4)

(4.1) L(s, π, r) =k∏i=1

di−1∏j=0

L(s+ j − di − 1

2, πi).

5The following discussion does not apply verbatim to certain non-connected groupschemes which are natural to consider here, such as On or PGOn. We will be more precisein the text about the necessary modifications needed to deal with them as well, but weshall ignore this detail in this introduction.

6As is customary, we denote by Πcusp(G) ⊂ Πdisc(G) the subset of representationsappearing in the subspace of cuspforms [GGPS66] (§IV.3).

11

In a slightly abusive way, the collection of conjugacy classes r(cv(π)) will becalled the Langlands parameter of (π, r), and we shall denote it by ψ(π, r).When the equality (4.1) holds, we shall also write it symbolically as follows7

ψ(π, r) = ⊕ki=1πi[di].

If G is a classical group over Z (§VI.4.7, §VIII.1), its dual group G isa complex classical group (that is special orthogonal, or symplectic). Inparticular, G possesses a natural or “tautological” representation, called thestandard representation, and denoted by St. An important result proved byArthur in [Art13] asserts that the Arthur-Langlands conjecture holds for(π, St) for all π in Πdisc(G) if G is either Sp2g or a split special orthogonalgroup over Z.

In Chapter VII, we shall illustrate these ideas by giving numerous exam-ples of specific cases of the Arthur-Langlands conjecture stated above, con-cerning automorphic forms for SOn or Siegel modular forms for Sp2g(Z).They do not rely on Arthur’s results, but rather on more classical construc-tions of theta series. We recall Rallis’ point of view on Eichler’s commu-tation relations (§VII.1) as well as important results of Böcherer [Boc89]and of Ikeda [Ike01]. We prove Theorem C and give other applications oftriality to the construction of certain elements in Πdisc(SO8) (§VII.2). Oneingredient of the proofs is a slight refinement of Rallis’ identities to the pair(PGOn,PGSp2g) (§VII.1.4). In fine, our analysis covers enough construc-tions to allow us to determine ψ(π, St) for 13 of the 16 “first” representationsπ in Πdisc(SO8) (§VII.4).

We are now able to state the analog for X24 of Theorem A; we refer to§X.1 for a formulation of the same theorem in terms of representations ofGal(Q/Q), in the spirit of our announcement [CL11]. We shall need a fewextra notations:

– The representation ∆w, for w ∈ {11, 15, 17, 19, 21}, will denote the ele-ment of Πcusp(PGL2) generated by the one dimensional space Sw+1(SL2(Z))of cuspforms of weight w + 1 for the group SL2(Z).

– The representation Sym2∆w is the Gelbart-Jacquet symmetric square of∆w [GJ78]. It is the unique element of Πcusp(PGL3) satisfying the equalitycv(Sym2∆w) = Sym2 cv(∆w) for all places v of Q.

7In the strict sense, we shall include in this equality the natural corresponding iden-tity at the Archimedean place (§VI.4.4). Moreover, the “summand” πi[di] will be simplydenoted by [di] (resp. πi) when ni = 1 (resp. di = 1). Those conventions are in force inTable I.2.

12

– Let (w, v) be one of the 4 couples (19, 7), (21, 5), (21, 9) or (21, 13).We will denote by ∆w,v a certain representation in Πcusp(PGL4) defined andstudied in §IX.1. The eigenvalues of its infinitesimal character c∞(∆w,v),which is by definition a semisimple conjugacy class in M4(C), are8 ±w

2and

±v2; we shall eventually show that this property characterizes ∆w,v uniquely.

Theorem E. The parameters ψ(π, St) of the 24 representations π in Πdisc(O24)generated by the maps X24 → C which are eigenvectors for H(O24) are :

[23]⊕ [1] Sym2∆11 ⊕∆17[4]⊕∆11[2]⊕ [9]

Sym2∆11 ⊕ [21] Sym2∆11 ⊕∆15[6]⊕ [9]

∆21[2]⊕ [1]⊕ [19] ∆15[8]⊕ [1]⊕ [7]

Sym2∆11 ⊕∆19[2]⊕ [17] ∆21[2]⊕∆17[2]⊕∆11[4]⊕ [1]⊕ [7]

∆21[2]⊕∆17[2]⊕ [1]⊕ [15] ∆19[4]⊕∆11[4]⊕ [1]⊕ [7]

∆19[4]⊕ [1]⊕ [15] ∆21,9[2]⊕∆15[4]⊕ [1]⊕ [7]

Sym2∆11 ⊕∆19[2]⊕∆15[2]⊕ [13] Sym2∆11 ⊕∆19[2]⊕∆11[6]⊕ [5]

Sym2∆11 ⊕∆17[4]⊕ [13] Sym2∆11 ⊕∆19,7[2]⊕∆15[2]⊕∆11[2]⊕ [5]

∆17[6]⊕ [1]⊕ [11] ∆21[2]⊕∆11[8]⊕ [1]⊕ [3]

∆21[2]⊕∆15[4]⊕ [1]⊕ [11] ∆21,5[2]⊕∆17[2]⊕∆11[4]⊕ [1]⊕ [3]

∆21,13[2]⊕∆17[2]⊕ [1]⊕ [11] Sym2∆11 ⊕∆11[10]⊕ [1]

Sym2∆11 ⊕∆19[2]⊕∆15[2]⊕∆11[2]⊕ [9] ∆11[12]

Table I.2: The standard parameters of the π ∈ Πdisc(O24) such that π∞ = 1.

Let us stress that in a remarkable work [Ike06], Ikeda had been able todetermine 20 of the 24 parameters above, namely the ones which do notcontain any representation of the form ∆w,v.

Given the importance of the role played by those ∆w,v in this memoir,let us say a bit more about their origin. Let (j, k) be one of the 4 couples(6, 8), (4, 10), (8, 8) or (12, 6). A dimension formula due to R. Tsushima showsthat the space of vector-valued Siegel modular cuspforms for Sp4(Z), withcoefficients in the representation SymjC2⊗ detk of GL2(C), has dimension 1[Tsu83]. We will give a concrete generator of this space using a constructionof theta series with “pluriharmonic” coefficients based of the E8 lattice. If πj,k

8In a similar way, the eigenvalues of c∞(∆w) are ±w2 .

13

denotes the element of Πcusp(PGSp4) generated by this eigenform, then wehave the relation ψ(πj,k, St) = ∆w,v with (w, v) = (2j+k−3, j+1). Observethat PGSp4 is isomorphic to the split classical group SO3,2 over Z, whosedual group is Sp4 over C, so that Arthur’s theory does apply to (πj,k, St).

Theorem E will be proved in §X.4.3, using a method that we shall describein §I.6. Nevertheless, we will first give two conditional proofs of this resultin §X.2.10 and §X.2.11. These proofs, obtained by applying Arthur’s multi-plicity formula, will eventually be the most natural ones, but they depend atthe moment on some refinements of Arthur’s results which are expected butnot yet available.

In Chapter VIII we thus come back to the general results of Arthur[Art13], that we specify to the case of classical groups over Z and of theirautomorphic forms which are “unramified everywhere”; such an analysis hadalready been mostly carried out in [CR12, §3], that we develop and complete.Most of Chapter VIII is devoted to make explicit the famous aforementionedmultiplicity formula. This formula gives a necessary and sufficient condi-tion on a given collection of (πi, di)

′s to “arise” as the standard parameter ofsome π in Πdisc(G) having furthermore a prescribed Archimedean componentπ∞ (§VIII.3); the first part of the assertion simply means that we ask theequality ψ(π, St) = ⊕ki=1πi[di]. We limit ourselves to the case where π∞ isa discrete series representation of G(R) and explain in concrete terms theway those representations are parameterized by Shelstad, which is the ingre-dient needed to understand Arthur’s formula (§VIII.4). At the moment, theversion of the formula that we give is only proved if G is split over Z andif all the integers di are equal to 1. Nevertheless, we discuss it in general,and we make precise the conjectures on which various specific cases depend(§VIII.4.21), because it greatly enlightens several constructions studied inthis memoir. In particular, we draw very concrete and explicit formulas inthe specific cases of Siegel modular forms for Sp2g(Z) and automorphic formsfor SOn (§VIII.5). We check that those formulas are compatible with the re-sults of Chapter VII and with the results of Böcherer on the image of theϑ maps (3.1) (§VIII.6). As promised, we show in §IX.2 that they lead to arather miraculous, but simple, conditional proof of Theorem E.

5. Algebraic automorphic representations of small weight

Let n ≥ 1 be an integer. We say that a representation π in Πcusp(PGLn) isalgebraic if the eigenvalues λi of c∞(π) satisfy λi ∈ 1

2Z and λi−λj ∈ Z for all

i, j. The largest difference between two eigenvalues of c∞(π) is then called the

14

motivic weight of π; it is a nonnegative integer denoted by w(π). As alreadyseen in the preface9, these algebraic cuspidal automorphic representationsare interesting in their own rights: they are precisely the ones which arerelated to `-adic “geometric” Galois representations according to the yoga ofFontaine-Mazur and Langlands (§VIII.2.16). The reason we are interested inthese representations here is a little different, and explained by the followingobservation.

Let G be a semisimple group scheme over Z, let π in Πdisc(G) be suchthat π∞ has the same infinitesimal character as an algebraic finite dimen-sional representation V of G(C), and let r : G(C)→ SLn(C) be an algebraicrepresentation. Assume that we have ψ(π, r) = ⊕ki=1πi[di] following Arthurand Langlands. Then the representations πi are algebraic and their motivicweight is bounded in terms of the highest weights of V and r (§VIII.2). Forinstance, if G = Sp2g and if π in Πcusp(Sp2g) is generated by a Siegel modulareigenform of weight k for Sp2g(Z) (say with k > g, but this condition can beweakened), then we may write ψ(π, St) = ⊕ki=1πi[di] thanks to Arthur, andeach πi is algebraic of motivic weight ≤ 2k − 2. An important ingredient inour proofs is the following classification statement, which is of independentinterest, and proved in §IX.3.

Theorem F. Let n ≥ 1 and let π in Πcusp(PGLn) be algebraic of motivicweight ≤ 22. Then π belongs to the following list of 11 representations :

1, ∆11, ∆15, ∆17, ∆19, ∆19,7, ∆21, ∆21,5, ∆21,9, ∆21,13, Sym2∆11.

In motivic weight < 11, this theorem states that we have n = 1 and thatπ is the trivial representation, a result which was already known to Mestreand Serre (up to the language, see [Mes86, §III, Remarque 1]). In this veryspecific case, it provides us (among other things!) with an “automorphic”analog of a classical theorem of Minkowski asserting that every number fielddifferent from Q contains at least one ramified prime (case w(π) = 0), andalso of a conjecture of Shaffarevic, independently proved by Abrashkin andFontaine, according to which there is no abelian variety over Z (case w(π) =1). As far as we know, the special case w(π) = 11 of the theorem is alreadynew. Let us emphasize that there is no assumption on n in the statement ofTheorem F, and that it implies n ≤ 4.

Our proof of Theorem F, in the spirit of the works of Stark, Odlyzko andSerre on lower bounds of discriminants of number fields, relies on an analogue

9The definition of algebraic given in the preface, apparently less restrictive, is actuallyequivalent to the one given above: see Remark VIII.2.14. The motivic weight w(π) is alsotwice the largest eigenvalue of c∞(π).

15

in the framework of automorphic L-functions of the so-called explicit formulasof Riemann and Weil in the theory of prime numbers. This analogue has beendeveloped by Mestre [Mes86] and applied by Fermigier to the standard L-function L(s, π) for π in Πcusp(PGLn) to show the nonexistence of certain π’s[Fer96]. We shall apply it more generally to the “Rankin-Selberg L-function”of an arbitrary pair {π, π′} of cuspidal automorphic representations of PGLnand PGLn′ (Jacquet, Piatetski-Shapiro, Shalika).

In the specific case where π′ is the dual of π, this method had already beensuccessfully employed by Miller [Mil02]; however our study contains somenovelties which deserve to be mentioned. First of all, we have discovered thatcertain symmetric bilinear forms, which are defined on the Grothendieck ringK∞ of the Weil group of R and real-valued, and which naturally occur ina formulation of the explicit formulas, are positive definite on rather largesubgroups of K∞. It is this phenomenon which is responsible for the finitenessof the list of the statement of Theorem F. Moreover, we establish some simplebut efficient criterions, for instance involving only π∞ and π′∞, that preventthe simultaneous existence of π and π′. We refer to §IX.3 for more precisestatements.

6. Proofs of Theorems D and E

Let us sketch the main steps in the proof of Theorem E (§IX.4.3). Letπ in Πdisc(O24) be such that π∞ is the trivial representation. The results ofErokhin and Borcherds-Freitag-Weissauer recalled in §I.3 show that π admitsa “ϑ-correspondent” π′ in Πcusp(Sp2g), generated by a Siegel modular form ofweight 12 and genus ≤ 11 for Sp2g(Z) (§VII.1), except for a single π satisfyingψ(π, St) = ∆11[12] already determined by Ikeda. Arthur’s theorem appliedto π′, and the point of view of Rallis on Eichler’s relations, imply that (π, St)satisfies the Arthur-Langlands conjecture. A simple combinatorial argument,relying only on Theorem F, shows that there are at most 24 possibilities forψ(π, St), namely the ones given in Table I.2. On the other hand, there areat least 24 possibilities for ψ(π, St), as the Hecke operator T2 has distincteigenvalues on C[X24] according to Nebe and Venkov: this concludes theproof.

This method allows to study more generally the elements of Πcusp(Sp2g)generated by a Siegel modular form of weight k ≤ 12 for the group Sp2g(Z)(§IX.5). The statement of Theorem D is the result of our study in the specialcase k = 12. We find exactly 23 Siegel modular forms for Sp2g(Z) which areeigenvectors for H(Sp2g), of weight 12 and genus g ≤ 23, and we even give

16

their standard parameters (Table C.1). In the case of forms of weight k ≤ 11,we prove the following theorem, generalizing previous results of [DI98] and[PY07] (Theorem IX.5.10).

Theorem G. Let g ≥ 1 be an integer and let k ∈ Z.

(i) If k ≤ 10 then Sk(Sp2g(Z)) vanishes, unless (k, g) is among

(8, 4), (10, 2), (10, 4), (10, 6), (10, 8),

in which case Sk(Sp2g(Z)) has dimension 1. The standard parameters ofthe 5 elements of Πcusp(Sp2g) generated by thoses spaces are respectively

∆11[4]⊕[1], ∆17[2]⊕[1], ∆15[4]⊕[1], ∆17[2]⊕∆11[4]⊕[1] and ∆11[8]⊕[1].

(ii) If k = 11 then Sk(Sp2g(Z)) vanishes, except maybe if g = 6.

Let us mention some difficulties in the proofs of Theorems D and G whichdo not appear in the one of Theorem E. Let π in Πcusp(Sp2g) be generatedby a Siegel modular form of weight k ≤ 12 and genus g < 12+k. Theorem Fallows to show that ψ(π, St) belongs to an explicit finite list of possibilities.Contrary to the situation of Theorem E, some elements of this list shouldactually not occur, as an inspection of the multiplicity formula shows. We by-pass the use of this formula by relying instead on results of Böcherer [Boc89],Ikeda [Ike01, Ike13], as well as on various theta series constructions. Weexpect that S11(Sp12(Z)) vanishes but we can’t prove it unconditionally. Thecases g ≥ k are more delicate (we don’t even know the explicit form ofArthur’s multiplicity formula in these cases); they are excluded in a ratherad hoc manner by using the work of S. Mizumoto [Miz91] on the poles of theL-function L(s, π, St) (§VIII.7).

7. Some applications

Thanks to Theorem E, the original problem of determining the numbersNp(L,M) for L,M in X24 and p prime becomes equivalent to the one of de-termining the eigenvalues of the Hecke operators in H(PGSp4) on the 4 genus2 vector-valued Siegel modular eigenforms mentioned in §I.4. We explain in§X.3 a method to compute those eigenvalues that we discovered, which relieson our analysis of the p-neighbors of the Leech lattice made in §III.4.

17

Let (j, k) be one of the 4 couples considered in §I.4, that is (6, 8), (4, 10),(8, 8) or (12, 6). Let (w, v) denote the corresponding couple (2j+k−3, j+1).If q is of the form pk where p is a prime and k an integer ≥ 1, we set

τj,k(q) := qw2 trace cp(∆w,v)

k.

This complex number is actually in Z.

Theorem H. Let (j, k) be one of the 4 couples (6, 8), (4, 10), (8, 8) or (12, 6).The integers τj,k(p) with p a prime ≤ 113, and τj,k(p2) with p a prime ≤ 29,are respectively given by the tables C.3 and C.4.

These results confirm and extend previous computations by Faber et vander Geer [FvdG04] [vdG08, §25] for p ≤ 37 by very different methods. Ourcomputations allow to determine the exact value of Np(L,M) for all L,M inX24 and all primes p ≤ 113.

Theorem F shows that the computation of τj,k(q) is perhaps not as fu-tile as it may seem. Indeed, it suggests a parallel classification, yet to beproved on the “`-adic side”, of the effective pure motives over Q, with goodreduction everywhere, whose motivic weight is ≤ 22. For instance, it im-poses a remarkable conjectural constraint on the Hasse-Weil zeta functionof the Deligne-Mumford stack Mg,n classifying the stable curves of genus gequipped with n marked points, say when g ≥ 2, n ≥ 0 and 3g− 3 +n ≤ 22:this zeta function should be expressible in terms of the τj,k(q) and of thecoefficients of the normalized cuspforms of weight ≤ 22 for SL2(Z). Thisconfirms certain results (resp. conjectures) of Bergström, Faber and van derGeer [FvdG04, Fab13, BFG14] when g = 2 (resp. g = 3).

In §X.4 we use Theorem E to prove some congruences satisfied by theintegers τj,k(p), say with p prime. They are obtained as a consequence of astudy of the eigenvectors of T2 in the natural basis of Z[X24] and by someGalois representations arguments. Among these congruences, we prove thefollowing one conjectured by Harder in [Har08].

Theorem I. (Harder’s conjecture) For each prime p we have the congruence

τ4,10(p) ≡ τ22(p) + p13 + p8 mod 41,

where τ22(p) denotes the le p-th coefficient of the normalized cuspform ofweight 22 for the group SL2(Z).

18

Let us go back to the proof of Theorem E sketched in §I.6. It relies onthe equality |X24| = 24, which is a consequence of Niemeier’s classification.However, we explain in §IX.6 how a combination of the ideas above and ofArthur’s multiplicity formula (including the conjectures 4.22 and 4.25 for-mulated in §VIII) allows to bypass the use of this equality, and even to givea new proof of it “without any Euclidean lattice computation”. Even bet-ter, we not only recover the fact that there are exactly 24 isometry classes ofNiemeier lattices, but also that there is a unique such lattice with no isometryof determinant −1.

Is it reasonnable to hope for a fine estimate of the cardinality of X32

using such a method ? The question remains open, but the example ofthe dimension 24 shows that this approach, dear to the first author, is nottotally absurd. A necessary ingredient in this project is the knowledge ofthe algebraic representations (say “selfdual, regular”) in Πcusp(PGLn) whosemotivic weight is ≤ 30: some progresses in this direction have recently beenobtained in [CR12] and [Tai14].

** *

To end this introduction, let us say a word about the use in this memoirof the recent results of Arthur. They rely on an impressive collection ofdifficult works, whose last pieces have been established only very recently(see [Art13], [MW14, Wal14], as well as the discussion in §VIII.1). We thusfound it useful to indicate throughout the memoir by a star ∗ the statementsthat depend on the results of Arthur’s book [Art13]. In this introduction,it concerns the proofs of Theorems B, D, E, F10, G, H and I. However, let usemphasize that contrary to our initial announcements in [CL11] and [Che13],our final proofs neither rely on the results announced by Arthur loc. cit. inhis last chapter 9 concerning inner forms, nor on the conjectural propertiesof Arthur’s packets of the type of the ones studied by Adams and Johnson.

10We actually prove a version of Theorem F which is almost as strong and which doesnot rely on Arthur’s results: see Theorem IX.3.2.

19

20